Iranian Journal of Fuzzy Systems Vol. 11, No. 5, (2014) pp. 39-46
39
SOME REMARKS ON GENERALIZED SEQUENCE SPACE OF
BOUNDED VARIATION OF SEQUENCES OF FUZZY NUMBERS
H. ALTINOK, M. ET AND R. ÇOLAK
Abstract. The idea of difference sequences of real (or complex) numbers was
introduced by Kızmaz [8]. In this paper, using the difference operator and a
lacunary sequence, we introduce and examine the class of sequence bvθ (∆, F ) .
We study some of its properties like solidity, symmetricity, etc.
1. Introduction
The theory of sequences of fuzzy numbers was first introduced by Matloka [10].
Matloka [10] introduced bounded and convergent sequences of fuzzy numbers, studied some of their properties and showed that every convergent sequence of fuzzy
numbers is bounded. Since then, there has been increasing interest in the study of
sequences of fuzzy numbers (see [3, 13, 14]). Lacunary sequences of fuzzy numbers
and real (or complex) numbers has been studied by Nuray [11], Kwon and Sung [9],
Altinok et al. [1], and many others [7, 12]. Recently, Başar et al. [4] and Tripathy
and Dutta [14] have studied the spaces of bounded variation sequences.
2. Preliminaries
In this section we give the basic notions related to fuzzy numbers and a brief
information about difference sequences and lacunary sequences
A fuzzy set u on R is called a fuzzy number if it has the following properties:
i) u is normal, that is, there exists x0 ∈ R such that u(x0 ) = 1;
ii) u is fuzzy convex, that is, for x, y ∈ R and 0 ≤ λ ≤ 1, u(λx + (1 − λ)y) ≥
min[u(x), u(y)];
iii) u is upper semicontinuous;
iv) supp u = cl{x ∈ R : u(x) > 0}, or denoted by [u]0 , is compact.
α−level set [u]α of a fuzzy number u is defined by
{x ∈ R : u(x) ≥ α}, if α ∈ (0, 1]
α
[u] =
supp u,
if α = 0.
It is clear that u is a fuzzy number if and only if [u]α is a closed interval for each
α ∈ [0, 1] and [u]1 6= ∅.
A real number r can be regarded as a fuzzy number r̄ defined by
1, x = r
r̄ (x) =
.
0, x 6= r
Received: February 2012; Revised: July 2012; Accepted: August 2014
Key words and phrases: Fuzzy number, Difference operator, Lacunary sequence.
40
H. Altinok, M. Et and R. Çolak
If u ∈ L(R), then u is called a fuzzy number, and L(R) is said to be a fuzzy number
space.
Let u, v ∈ L(R), k ∈ R and the α−level sets of fuzzy numbers u and v be
α
α
[u] = [uα , uα ] and [v] = [v α , v α ] , α ∈ (0, 1] . Then, a partial ordering ” ≤ ”
in L(R) is defined by u ≤ v ⇔ uα ≤ v α and uα ≤ v α for all α ∈ (0, 1] . Some
arithmetic operations for α−level sets are defined as follows:
α
[u + v] = [uα + v α , uα + v α ]
α
[u − v] = [uα − v α , uα − v α ]
[kuα , kuα ] , if k ≥ 0
.
[ku]α =
[kuα , kuα ] , otherwise
In order to calculate the distance between two fuzzy numbers u and v, we use
the metric
α
α
d (u, v) = sup dH ([u] , [v] )
0≤α≤1
where dH is the Hausdorff metric defined as
α
α
dH ([u] , [v] ) = max {|uα − v α | , |uα − v α |} .
It is known that d is a metric on L(R), and (L(R), d) is a complete metric space.
A sequence X = (Xk ) of fuzzy numbers is a function X from the set N of all positive integers into L(R). Thus, a sequence of fuzzy numbers (Xk ) is a correspondence
from the set of positive integers to a set of fuzzy numbers.
Let X = (Xk ) be a sequence of fuzzy numbers. The sequence X = (Xk ) of fuzzy
numbers is said to be bounded if the set {Xk : k ∈ N} of fuzzy numbers is bounded
and convergent to the fuzzy number X0 , written as lim Xk = X0 , if for every ε > 0
k
there exists a positive integer k0 such that d (Xk , X0 ) < ε for k > k0 . Let `∞ (F )
and c (F ) denote the set of all bounded sequences and all convergent sequences of
fuzzy numbers, respectively [10].
The difference spaces `∞ (∆), c (∆) and c0 (∆), consisting of all real valued
sequences x = (xk ) such that ∆x = ∆1 x = (xk − xk+1 ) in the sequence spaces
`∞ , c and c0 , were defined by Kızmaz [8]. The idea of difference sequences was
generalized by Et and Çolak [6], Et et al. [5], Altinok et al. [2] and Tripathy et al.
[15–17].
Let w(F ) be the set of all sequences of fuzzy numbers. The operator ∆ : w(F ) →
w(F ) is defined by ∆Xk = Xk − Xk+1 .
A sequence space E(F ) is said to be normal (or solid) if (Xk ) ∈ E(F ) and (Yk )
is such that d (Yk , 0̄) ≤ d (Xk , 0̄) implies (Yk ) ∈ E(F ).
A sequence space E(F ) is said to be symmetric if Xπ(n) ∈ E(F ), whenever
(Xk ) ∈ E(F ), where π is a permutation of N.
A sequence space E(F ) is said to be convergence free if (Yk ) ∈ E(F ), whenever
(Xk ) ∈ E(F ) and Xk = 0̄ implies Yk = 0̄.
By a lacunary sequence θ = (kr ) ; r = 0, 1, 2, ..., where k0 = 0, we mean an
increasing sequence of non-negative integers with hr = (kr − kr−1 ) → ∞ as r → ∞.
kr
.
The intervals determined by θ will be denoted by Ir = (kr−1 , kr ] and qr = kr−1
Some Remarks on Generalized Sequence Space of Bounded Variation of ...
41
P∞
A sequence (xk ) is said to be of bounded variation if k=1 |∆xk | < ∞. It is
denoted by
(
)
∞
X
bv = (xk ) ∈ w :
|∆xk | < ∞ ,
k=1
where ∆xk = xk − xk+1 , for all k ∈ N. The class of fuzzy real valued bounded
variation sequences bv (F ) is given by
(
)
∞
X
bv (F ) = (Xk ) ∈ w (F ) :
d (Xk , Xk+1 ) < ∞ .
k=1
3. Main Results
In this section, we introduce the class of sequence bvθ (∆, F) and give an inclusion relation. In addition, we show that this class of sequence is neither solid nor
symmetric nor convergence free.
We define the class of sequence bvθ (∆, F ) as follows:
!
)
(
∞
X
1 X
d (∆Xk , ∆Xk+1 ) < ∞ .
bvθ (∆, F ) = X = (Xk ) ∈ w(F ) :
hr
r=1
k∈Ir
r
In the special case θ = 2 , we shall write bv (∆, F ) instead of bvθ (∆, F ) .
Theorem 3.1. The class of sequence bvθ (∆, F) is closed under the operations of
addition and scalar multiplication.
Proof. (i) Let (Xk ) and (Yk ) be two sequences of fuzzy numbers in bvθ (∆, F) and
θ = (kr ) be a lacunary sequence. Then we can write the following inequality:
!
∞
X
1 X
d [(∆Xk + ∆Yk ) , (∆Xk+1 + ∆Yk+1 )]
hr
r=1
k∈Ir
!
!
∞
∞
X
X
1 X
1 X
≤
d (∆Xk , ∆Xk+1 ) +
d (∆Yk , ∆Yk+1 )
hr
hr
r=1
r=1
k∈Ir
k∈Ir
Since (Xk ) , (Yk ) ∈ bvθ (∆, F) , the right side of inequality is less than infinity, so
the left side is. Hence bvθ (∆, F) is closed under the operation of addition.
(ii) Let (Xk ) be a sequence of fuzzy numbers in bvθ (∆, F) and θ = (kr ) be a
lacunary sequence and c ∈ R. Then we can write
!
!
∞
∞
X
X
1 X
1 X
d (c∆Xk , c∆Xk+1 ) = |c|
d (∆Xk , ∆Xk+1 ) < ∞.
hr
hr
r=1
r=1
k∈Ir
k∈Ir
Since (Xk ) ∈ bvθ (∆, F) , the right side of inequality is less than infinity. Therefore
we obtain (c∆Xk ) ∈ bvθ (∆, F) and so bvθ (∆, F) is closed under the operation of
scalar multiplication.
P∞ 1 Theorem 3.2. Let θ be a lacunary sequence such that hr 6= r. If r=1 hr < ∞,
then bv (F) ⊂ bv (∆, F) ⊂ bvθ (∆, F).
42
H. Altinok, M. Et and R. Çolak
Proof. Let (Xk ) be a sequence of fuzzy numbers and (Xk ) ∈ bv (F) . Then we have
∞
P
d (Xk , Xk+1 ) < ∞. From the definition of difference operator and metric d, we
k=1
can write
∞
X
d (∆Xk , ∆Xk+1 ) =
k=1
≤
∞
X
k=1
∞
X
d (Xk − Xk+1 , Xk+1 − Xk+2 )
d (Xk , Xk+1 ) +
k=1
∞
X
d (Xk+1 , Xk+2 ) < ∞.
k=1
Hence bv (F) ⊂ bv (∆, F) .
POn the other hand, we can find a positive integer N
d (∆Xk , ∆Xk+1 ) < 1, for all r ≥ N. Therefore, we can write
such that
k∈Ir
X
r≥N
!
X 1
1 X
d (∆Xk , ∆Xk+1 ) ≤
<∞
hr
hr
r≥N
k∈Ir
and so
∞
X
r=1
!
1 X
d (∆Xk , ∆Xk+1 ) < ∞.
hr
k∈Ir
Thus we have (Xk ) ∈ bvθ (∆, F) .
Theorem 3.3. The class of sequence bvθ (∆, F) is not solid.
Proof. To prove the theorem it is sufficient to give the following example.
Example 3.4. Let (Xk ) be a sequence of fuzzy numbers and θ = (5r ) be a lacunary
sequence. Define the sequence (Xk ) of fuzzy numbers by
 k
 2 t + 1, − k2 ≤ t ≤ 0
Xk (t) =
− k t + 1, 0 ≤ t ≤ k2 .
 2
0,
otherwise
Using the membership function of (Xk ) , we calculate the α−level sets of (Xk ) and
(∆Xk ) , respectively, as
2α − 2 2 − 2α
α
[Xk ] =
,
k
k
and
(2α − 2) (2k + 1) (2 − 2α) (2k + 1)
α
[∆Xk ] =
,
k (k + 1)
k (k + 1)
for α ∈ [0, 1] and obtain the membership function of (∆Xk ) as
∆Xk (t) =





k(k+1)
4k+2 t + 1,
1 − k(k+1)
4k+2 t,
0,
4k+2
− k(k+1)
≤t≤0
4k+2
0 ≤ t ≤ k(k+1)
otherwise
Some Remarks on Generalized Sequence Space of Bounded Variation of ...
43
Then we get
∞
X
!
∞
X
1 X
d (∆Xk , ∆Xk+1 ) =
hr
r=1
r=1
k∈Ir
X
4
1
r−1
4.5
k (k + 2)
!
< ∞.
k∈Ir
Hence (Xk ) ∈ bvθ (∆, F) .
On the other hand, define the sequence (Yk ) by
α
Xk , for k odd
Yk =
.
0̄, for k even
h
i
α
where [Xk ] = Xkα , Xkα . That is, (Yk ) = X1α , 0, X3α , 0, X5α , 0, X7α , 0, ... and so
we have difference sequence of (Yk ) as
(∆Yk ) = X1α , −X3α , X3α , −X5α , X5α , −X7α , X7α , ... .
Then we get
∞
X
r=1
1 X
d (∆Yk , ∆Yk+1 )
hr
k∈Ir
!
>
∞
X
r=1
X 2
1
4.5r−1
k
!
= ∞.
k∈Ir
Therefore it implies that (Yk ) ∈
/ bvθ (∆, F) from the definition.
Theorem 3.5. The class of sequence bvθ (∆, F) is not symmetric.
Proof. We give the following example to show that bvθ (∆, F) is not symmetric.
Example 3.6. Let (Xk ) be a sequence of fuzzy numbers and θ = (5r ) be a lacunary
sequence. Define the sequence (Xk ) of fuzzy numbers by


t + 2, −2 ≤ t ≤ 0 


if k = 5m

2 − t, 0 ≤ t ≤ 2
:= L,
m∈N
.
Xk (t) =

0,
otherwise



0̄,
otherwise
Then we calculate the α−level sets of (Xk ) and (∆Xk ) , respectively, as
[α − 2, 2 − α] , if k = 5m
α
[Xk ] =
.
0̄,
otherwise
and
[α − 2, 2 − α] , if k = 5m and k + 1 = 5m
0̄,
otherwise
α
for α ∈ [0, 1] . That is, we write the difference sequence ([∆Xk ] ) = (Lα , 0̄, 0̄, Lα , Lα ,
α
α
α
0̄, ..., 0̄, L , L , 0̄, ...) , where L = [α − 2, 2 − α] . Then Xk ∈ bvθ (∆, F) is clear
from the following inequalities:
!
!
∞
∞
X
X
X
1 X
1
d (∆Xk , ∆Xk+1 ) ≤
2
hr
4.5r−1
r=1
r=1
k∈Ir
k∈Ir
∞ X
4
≤
< ∞.
4.5r−1
r=1
α
[∆Xk ] =
44
H. Altinok, M. Et and R. Çolak
Now, we define a sequence (Yk ) which is the arrangement of the sequence (Xk )
by
(Yk ) = (L, 0, 0, L, 0, 0, L, 0, 0, L, 0, 0, L, ...)
and calculate the difference sequence as
(∆Yk ) = (L, 0, L, L, 0, L, L, 0, L, L, 0, L, L, ...) .
Then
∞
X
r=1
1 X
d (∆Yk , ∆Yk+1 )
hr
k∈Ir
!
=
∞ X
r=1
1
4.5r−1
r−1 X
∞
4.5
4. +2
>
1 = ∞.
3 r=1
Hence we get Yk ∈
/ bvθ (∆, F) . This shows that bvθ (∆, F) is not symmetric.
Theorem 3.7. The sequence space bvθ (∆, F) is not convergence free.
Proof. The proof follows from the following example.
Example 3.8. Let (Xk ) be a sequence of fuzzy numbers and θ = (5r ) be a lacunary
sequence. Consider the sequence (Xk ) of fuzzy numbers defined as follows:


k2
t + 1, − k22 ≤ t ≤ 0 

2


2
− k2 t + 1, 0 ≤ t ≤ k22  , if k is odd .
Xk (t) =

0,
otherwise


0̄,
if k is even
Then we obtain α−level sets of (Xk ) and (∆Xk ) as follows
( h
i
2(α−1) 2(1−α)
, k2
, if k is odd
α
2
k
.
[Xk ] =
0̄,
if k is even
and
 h
i
2(1−α)
 2(α−1)
,
, if k is odd
2
2
k
α
i
h k
[∆Xk ] =
2(1−α)
2(α−1)

,
, if k is even
(k+1)2 (k+1)2
for α ∈ [0, 1] and obtain the membership function of (∆Xk ) as


k2

t + 1, − k22 ≤ t ≤ 0 

2

2


if k is odd
1 − k2 t, 0 ≤ t ≤ k22 




0,
otherwise

∆Xk (t) =
(k+1)2
2
t
+
1,
−

2 ≤ t ≤ 0 

2

(k+1)


(k+1)2
2

if k is even

1
−
t,
0
≤
t
≤

2

(k+1)2



0,
otherwise
Then we have Xk ∈ bvθ (∆, F) from the following inequality:
!
!!
r−1 ∞
X
4.5
2
1 X
1
1
d (∆Xk , ∆Xk+1 ) =
−
< ∞.
2
hr
4.5r−1 2 k2
(k + 1)
r=1
r=1
k∈Ir
∞
X
Some Remarks on Generalized Sequence Space of Bounded Variation of ...
45
Now, we consider a sequence (Yk ) of fuzzy numbers defined by
 √

k

+ 1, − √2k ≤ t ≤ 0 


2 t√


k
, if k is odd
√2
1
−
t,
0
≤
t
≤
2
Yk (t) =
.
k 



0,
otherwise


0̄,
if k is even
To show Yk ∈
/ bvθ (∆, F) , firstly, we calculate α−level sets of (Yk ) and (∆Yk ) as
follows:
i
( h
2(α−1) 2(1−α)
√
, √k
, if k is odd
α
k
[Yk ] =
0̄,
if k is even
and
 h
i
2(1−α)
 2(α−1)
√
√
,
, if k is odd
α
k
k i
h
[∆Yk ] =
.
 2(α−1)
√
√
, 2(1−α)
, if k is even
k+1
k+1
for α ∈ [0, 1] . From this, we obtain the membership function of (∆Yk ) as follows:
√


k
t + 1, − √2k ≤ t ≤ 0 



2
√


k

if k is odd
√2

1
−
t,
0
≤
t
≤

2
k 



0,
otherwise
√

∆Yk (t) =
k+1

√2
≤t≤0 
t
+
1,
−


2 √
k+1



k+1

if k is even
√2
1
−
t,
0
≤
t
≤

2

k+1 


0,
otherwise
We can write
!
∞ X
1 X
2
d (∆Yk , ∆Yk+1 ) =
hr
4.5r−1
r=1
r=1
∞
X
k∈Ir
r−1 4.5
1
1
√ −√
= ∞.
2 k+1
k
Thus it implies that (Yk ) ∈
/ bvθ (∆, F) from the definition.
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H. Altinok∗ , Department of Mathematics, Firat University, Elazig, Turkey
E-mail address: [email protected]
M. Et, Department of Mathematics, Firat University, Elazig, Turkey
E-mail address: [email protected]
R. Çolak, Department of Mathematics, Firat University, Elazig, Turkey
E-mail address: [email protected]
*Corresponding author